What is the statement of the Stokes theorem

Stokes' theorem

In this article, the Stokes' theorem treated. First the general Stokes' theorem formulated before briefly on its special cases Main theorem of differential and integral calculus (HDI) as well as the Gaussian integral theorem is received. In addition, the classic integral theorem from Stokes as a further special case of the general will be examined in more detail. Finally, the calculation takes place two examples.

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  • Stokes general integral theorem
    in the text
  • Underlying topological principle
    in the text
  • Stokes' theorem as a classical Stokes integral theorem
    in the text
  • in the text

Stokes general integral theorem

If from Stokes' theorem is in question, in most cases it is the classic Stokes integral theorem meant. It represents a special case of the general integral theorem of Stokes which reads as follows:

Be open and a oriented -dimensional submanifold With as a continuously differentiable -Shape in . Then applies to everyone compact amount with a smooth edge


in which the induced orientation carries and the outer derivative of designated.

Underlying topological principle

The Stokes' theorem lies that topological principle is based on the fact that when paving a piece of land by identically oriented "paving stones" the inner ways in opposite direction will go through, which leads to their Mutually cancel contributions to the line integral and only that Contribution of the boundary curve remains.

Main theorem of differential and integral calculus as a special case

For degenerate the general integral theorem of Stokes to the Law of differential and integral calculus: Be an open interval and a continuously differentiable function. Then the following applies:

Gauss integral theorem as a special case

Another special case follows from the general integral theorem of Stokes the Gaussian integral theorem. To show that will be chosen and be it


i.e. with the continuously differentiable vector field . The roof shows over indicates that this factor must be omitted. Be also the outer unit normal field, then applies

With also results

Ultimately, this gives that Gaussian integral theorem

Stokes' theorem as a classical Stokes integral theorem

Often, and especially in technical courses and physics, there is talk of Stokes' theorem. This is usually the classic integral theorem from Stokes meant which one too Kelvin-Stokes theorem or Rotation set is called. Together with the Gaussian integral theorem he plays an essential role in the formulation of the Maxwell's equations in the integral form.

Special case of Stokes' general integral theorem

The classic theorem of Stokes results like that HDI and the Gaussian integral theorem as Special case of Stokes' general integral theorem. In this case the open amount as well as that continuously differentiable vector field considered. put one two-dimensional submanifold whose orientation through the Unit normal field is given. On the submanifold be on Compact form given which one smooth edge own. This in turn is through the Unit tangent field oriented. With the in continuously differentiable Pfaffian form


thus results in the Stokes' theorem:

In another notation it reads:

Stokes' theorem

The following can be seen:

The Stokes' theorem states that a Area integral about the rotation one Vector field under certain conditions in a closed curve integral over to the curve tangential component of Vector field can be converted. The curve traversed must be the Edge of the area under consideration correspond.

Stokes' theorem

In the following, the Stokes' theorem be proven. For this proof, however, there is a small one condition to the surface posed. This should be the graph a function be which over one area in the Level is defined. With and be the Projections of and the counterclockwise edge on the -Level. be through

parameterized, from which with the help of the Chain rule follows:

For the im Stokes' theorem looked at Curve integral:

By summarizing it results:

Will now